[AccessD] Weekend fun: Primes

Euclid did not prove that there are infinitely many primes. His alleged
proof requires the assumption that there are infitinitely many numbers,
which is an axiom (i.e. a religious act of faith rather than something that
can be demonstrated; or to put it another way, what is the sum of infinity -
1 plus infinity -1?). Axioms by definition are those things which appear
self-evident but cannot be proved. Witness the grand colossal failure of the
Russell-Whitehead Principia. We simply cannot prove such simple things as:
x + y = y + x
Because we cannot test all the possibilities. We assume its correctness, and
box that assumption in "axiom", but that hardly constitutes proof. We
sidestep such problems by calling them axioms.
Similarly, no one has got around Zeno's paradoxes in a couple of thousand
years, save for those who postulate that the universe is an involuted
hypersphere in which motion is a non-concept... but that can neither be
proved or disproved (yet). We simply trust that there exists an infinity of
numbers, and as Kantor demonstrated, we also need transfinite numbers. (One
might point out that we need an infinity of transfinite orders of magnitute,
but to understand recursion you must first understand recursion.)

-----Original Message-----
From: accessd-bounces at databaseadvisors.com
[mailto:accessd-bounces at databaseadvisors.com] On Behalf Of Stuart McLachlan
Sent: December 16, 2005 8:24 AM
To: Access Developers discussion and problem solving
Subject: Re: [AccessD] Weekend fun: Primes

http://primes.utm.edu/infinity.shtml
<quote>
About 2000 years ago Euclid proved that there were infinitely many primes. 
For mathematicians "infinity many" is an incomplete answer--they then ask 
"how big of an infinity?"  The prime number theorem, which states the 
number of primes less than x is approximately x/log x (the natural log), 
gives perhaps the best answer. 
......
the sum of the reciprocals of the primes diverges, so the primes are a 
"large" subset of the integers
</quote>
-- 
Stuart


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